2. Is there any way to run Stata with graphical interface remotely,
such as over X11 or another Mac program that lets you use a desktop
remotely?

First off, to the original poster, Stata for Mac has this annoying
habit of popping up in the Dock with a status bar when run in batch
mode. This is in effect a copy protection scheme, as only the user
currently logged into the Mac desktop (ie physically logged in) can
execute Stata jobs. If everyone shares an account on the Stata machine,
you are fine. But if some other user logs in locally, you will not be
able to run remote Stata jobs from the command line.

Yes. VNC is the protocol you want to look at.

I can recommend VNC as well. It's not a cutting edge innovation, but it
is free and cross-platform. Apple has an around $300 solution called
Remote Desktop, or something like that.

For the OSX server machine you will want to use either osxvnc
or "share my desktop"

In my experience, osxvnc is stabler than "share my desktop".

The advantage to this version is that you can tunnel/encrypt your
connection
over ssh.

Tunneling (port forwarding) is a feature of the ssh client, not the VNC
server or client being used. However, it is the case that osxvnc can be
setup to ONLY allow access from ssh clients doing port forwarding.
Read the documentation of osxvnc.

Perhaps what you mean is that some VNC clients automate the process for
you, and others require you to manually log in via a ssh client. That
may be correct, I do not know.

There is one instance where you probably don't want to use VNC as the
server
and that is on an XP desktop. While possible, it's probably better to
use
the native protocol from microsoft. You can access an XP machine from
OSX
using remote desktop client.

Yeah, the Windows NT family's Terminal Services architecture is very
well done, even though it triggers Microsoft's various licensing
schemes. On the low end, a hobbled version of Terminal Services is
included in Windows XP Pro (not Home) to allow a remote user to log in
when the machine is not in use by local user. In other words, logging
in remotely pauses the interactive session. There is a rumor that XP
service pack 2 was going to allow simultaneous remote and local users,
although I don't know if the rumor is accurate.